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Parikh vectors
8PAM_1 9CVD_1 5KCV_1 Letter Amino acid
32 9 26 A Alanine
10 10 28 R Arginine
9 15 14 N Asparagine
5 6 16 Q Glutamine
23 7 18 S Serine
9 8 18 Y Tyrosine
13 13 24 D Aspartic acid
1 16 6 C Cysteine
24 17 31 G Glycine
11 14 18 I Isoleucine
15 13 18 P Proline
3 2 7 W Tryptophan
30 10 27 V Valine
18 17 45 E Glutamic acid
11 6 18 H Histidine
25 11 42 L Leucine
6 18 36 K Lycine
3 5 15 M Methionine
4 10 26 F Phenylalanine
14 13 27 T Threonine 

8PAM_1|Chain A|Beta-lactamase VIM-1|Pseudomonas aeruginosa (287)
>9CVD_1|Chain A|Histone-lysine N-methyltransferase NSD2|Homo sapiens (9606)
>5KCV_1|Chain A|RAC-alpha serine/threonine-protein kinase|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
8PAM , Knot 117 266 0.81 40 152 253 
MLKVISSLLVYMTASVMAVASPLAHSGEPSGEYPTVNEIPVGEVRLYQIADGVWSHIATQSFDGAVYPSNGLIVRDGDELLLIDTAWGAKNTAALLAEIEKQIGLPVTRAVSTHFHDDRVGGVDVLRAAGVATYASPSTRRLAEAEGNEIPTHSLEGLSSSGDAVRFGPVELFYPGAAHSTDNLVVYVPSANVLYGGCAVHELSSTSAGNVADADLAEWPTSVERIQKHYPEAEVVIPGHGLPGGLDLLQHTANVVKAHKNRSVAE
9CVD , Knot 106 220 0.86 40 160 216 
SERKPPPYKHIKVNKPYGKVQIYTADISEIPKCNCKPTDENPCGFDSECLNRMLMFECHPQVCPAGEFCQNQCFTKRQYPETKIIKTDGKGWGLVAKRDIRKGEFVNEYVGELIDEEECMARIKHAHENDITHFYMLTIDKDRIIDAGPKGNYSRFMNHSCQPNCETLKWTVNGDTRVGLFAVCDIPAGTELTFNYNLDCLGNEKTVCRCGASNCSGFLG
5KCV , Knot 192 460 0.85 40 244 436 
MGHHHHHHENLYFQGSDVAIVKEGWLHKRGEYIKTWRPRYFLLKNDGTFIGYKERPQDVDQREAPLNNFSVAQCQLMKTERPRPNTFIIRCLQWTTVIERTFHVETPEEREEWTTAIQTVADGLKKQAAAEMDFRSGSPSDNSGAEEMEVSLAKPKHRVTMNEFEYLKLLGKGTFGKVILVKEKATGRYYAMKILKKEVIVAKDEVAHTLTENRVLQNSRHPFLTALKYSFQTHDRLCFVMEYANGGELFFHLSRERVFSEDRARFYGAEIVSALDYLHSEKNVVYRDLKLENLMLDKDGHIKITDFGLCKEGIKDGATMKTFCGTPEYLAPEVLEDNDYGRAVDWWGLGVVMYEMMCGRLPFYNQDHEKLFELILMEEIRFPRTLGPEAKSLLSGLLKKDPKQRLGGGSEDAKEIMQHRFFAGIVWQHVYEKKLSPPFKPQVTSETDTRYFDEEFTAQM

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(8PAM_1)}(2) \setminus P_{f(9CVD_1)}(2)|=86\), \(|P_{f(9CVD_1)}(2) \setminus P_{f(8PAM_1)}(2)|=94\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:11011001110101011111011100101010010100111101010011011100110001011101001111001001111001111000111110100011111001100010000111101101111100101000011010100110001011000101101111011011110000011101101011011011001000011011010110110010010000101011111011111101100010110100000110
Pair \(Z_2\) Length of longest common subsequence
8PAM_1,9CVD_1 180 3
8PAM_1,5KCV_1 178 3
9CVD_1,5KCV_1 196 3

Newick tree

 
[
	9CVD_1:95.72,
	[
		8PAM_1:89,5KCV_1:89
	]:6.72
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{486 }{\log_{20} 486}-\frac{220}{\log_{20}220})=76.9\)
Status Protein1 Protein2 d d1/2
Query variables 8PAM_1 9CVD_1 96 91.5
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]